Branch and Bound Hybrid Algorithm to Determine the Exact or Approximate Solution of the 0/1 Knapsack Problem with One Parameter

In this paper, the Zavala-Cruz algorithm is presented to solve the knapsack problem with one parameter. The foundation of the algorithm is shown; it is based largely on the algorithm of Horowitz-Sahni, with the lower and upper bounds taken from Dantzig, and the artifices of simulated annealing used to in order to escape of the local optimums. The Zavala-Cruz algorithm defines the search space and the rules to branch and prune, with which avoids the backtracking and this accelerates quicker the convergence to an exact or approximate solution. The Zavala-Cruz algorithm determines the exact solution for all the uncorrelated instances and for some of the weakly correlated instances. It also determines the approximate solution for the strongly correlated instances. These solutions were obtained along the whole parameter, including the transition phase, where the most complex instances are found, and where in some cases the instances are not computable. For each instance and for each value of the parameter, a multiple of n² was used in order to branch and prune, obtaining convergence with kn² iterations in all cases.